Excess Wealth Order Research Articles

Aging notions, stochastic orders, and expected utilities

AbstractThere are some connections between aging notions, stochastic orders, and expected utilities. It is known that the DRHR (decreasing reversed hazard rate) aging notion can be characterized via the comparative statics result of risk aversion, and that the location-independent riskier order preserves monotonicity between risk premium and the Arrow–Pratt measure of risk aversion, and that the dispersive order preserves this monotonicity for the larger class of increasing utilities. Here, the aging notions ILR (increasing likelihood ratio), IFR (increasing failure rate), IGLR (increasing generalized likelihood ratio), and IGFR (increasing generalized failure rate) are characterized in terms of expected utilities. Based on these observations, we recover the closure properties of ILR, IFR, and DRHR under convolution, and of IGLR and IGFR under product, and investigate the closure properties of the dispersive order, location-independent riskier order, excess wealth order, the total time on test transform order under convolution, and the star order under product. We have some new findings.

Some Results on Measures of Interaction among Risks

It has become a common understanding that financial risk can spread rapidly from one institution to another, and the stressful status of one institution may finally result in a systemic crisis. One popular method to assess and quantify the risk of contagion is employing the co-risk measures and risk contribution measures. It is interesting and important to understand how the underlining dependence structure and magnitude of random risks jointly affect systemic risk measures. In this paper, we mainly focus on the conditional value-at-risk, conditional expected shortfall, the delta conditional value-at-risk, and the delta conditional expected shortfall. Existing studies mainly focus on the situation with two random risks, and this paper makes some contributions by considering the scenario with possibly more than two random risks. By employing the tools of stochastic order, positive dependence concepts and arrangement monotonicity, several results concerning the usual stochastic order, increasing convex order, dispersive order and excess wealth order are presented. Concisely speaking, it is found that for a large enough stress level, a larger random risk tends to lead to a more severe systemic risk. We also performed some Monte Carlo experiments as illustrations for the theoretical findings.

On the second-order excess wealth order and its properties

AbstractIn the literature, some stochastic orders have been extended to the higher orders in different scenarios. In this paper, inspired by interesting properties of the excess wealth order and its wide range application particularly in comparing the tail variability of risks, we consider the second-order excess wealth order and study its main properties. We obtain two results characterizing the proposed order. We also investigate its relationship with other well-known variability orders and criteria to compare risks. An application of the results in comparing the epoch times of two nonhomogeneous poisson processes is also given.

What Determines the Information Rent under Reserve Prices: Equivalence with the Excess Wealth Order

What Determines the Information Rent under Reserve Prices: Equivalence with the Excess Wealth Order

Systemic risk: Conditional distortion risk measures

In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution (ΔCoD) measures as measures of systemic risk and analyze their properties and representations. The classes unify, and significantly extend, existing systemic risk measures such as the conditional Value-at-Risk, conditional Expected Shortfall, and risk contribution measures in terms of the VaR and ES. We provide sufficient conditions for two random vectors to be ordered by the proposed CoD-risk measures and ΔCoD-measures. These conditions are expressed using the conventional stochastic dominance, increasing convex/concave, dispersive, and excess wealth orders for the marginals and canonical positive/negative stochastic dependence notions.

Preservation of some stochastic orders by distortion functions with application to coherent systems with exchangeable components

AbstractThe preservation of stochastic orders by distortion functions has become a topic of increasing interest in the reliability analysis of coherent systems. The reason of this interest is that the reliability function of a coherent system with identically distributed components can be represented as a distortion function of the common reliability function of the components. In this framework, we study the preservation of the excess wealth order, the total time on test transform order, the decreasing mean residual live order, and the quantile mean inactivity time order by distortion functions. The results are applied to study the preservation of these stochastic orders under the formation of coherent systems with exchangeable components.

Generalized Entropies, Variance and Applications.

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

Stochastic orders and co-risk measures under positive dependence

Conditional risk measures (or co-risk measures) and risk contribution measures are increasingly used in actuarial portfolio analysis to evaluate the systemic risk, which is related to the risk that the failure or loss of a component spreads to another component or even to the whole portfolio: while co-risk measures are risk-adjusted versions of measures usually employed to assess isolate risks, risk contribution measures quantify how a stress situation for a component affects another one. In this paper, we provide sufficient conditions under which two random vectors could be compared in terms of CoVaR (conditional value-at-risk), CoES (conditional expected shortfall) and different risk contribution measures. Conditions are given in terms of the increasing convex order, the dispersive order and the excess wealth order of the marginals under some assumptions of positive dependence.

Stochastic comparisons of interfailure times under a relevation replacement policy

AbstractWe provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

On sufficient conditions for the comparison in the excess wealth order and spacings

Abstract The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

Variability ordering of multiplicative frailty models

The classical multiplicative frailty model in survival analysis accounts for unobserved heterogeneity between individuals. It is of great importance to identify how the variation of the frailty variable affects that of the overall population. This paper is mainly to present how the dispersive and the excess wealth orders between two frailty variables, translate into the corresponding orders between the resulting overall population variables. For the mean residual life and the mean inactivity time orders, we also obtain relevant analogous results in multiplicative frailty models.

The M/D/1+D queue has the minimum loss probability among M/G/1+G queues

We consider the loss probability Ploss in the stationary M/G/1 queue with generally distributed impatience times (M/G/1+G queue). Recently, it was shown that Ploss increases with service times in the convex order. In this paper, we show that Ploss also increases with impatience times in the excess wealth order. With these results, we show that Ploss in the M/D/1+D queue is smallest among all M/G/1+G queues with the same and finite arrival rate, mean service time, and mean impatience time.

Formula omitted]-metric under the location-independent risk ordering of random variables

The Lp-metric Δh,p(X) between the survival function F¯ of a random variable X and its distortion h∘F¯ is a characteristic of the variability of X. In this paper, it is shown that if a random variable X is larger than another random variable Y in the location-independent risk order or in the excess wealth order, then Δh,p(X)≥Δh,p(Y) whenever p∈(0,1] and the distortion function h is convex or concave. An alternative and simple proof of the corresponding known result in the literature for the dispersive order is given. Some applications are also presented.

Some results for the comparison of generalized order statistics in the total time on test and excess wealth orders

This paper is devoted to the study of the comparison of generalized order statistics in terms of the total time on test transform and the excess wealth orders. We provide some extensions of previous results in the literature for usual order statistics and generalized order statistics. These results involve results for the minimum of a random vector of generalized order statistics.

NEW PROPERTIES OF THE TOTAL TIME ON TEST TRANSFORM ORDER

The total time on test transform (ttt) order was introduced in Kochar, Li, and Shaked (2002) for nonnegative random variables, which has a close connection to the location independent riskier and the excess wealth orders. In this article, the ttt order is redefined for comparing not necessarily nonnegative random variables. Such an extension will lead us to conveniently derive several new intrinsic properties of the ttt order. In particular, we establish an interesting separation result on the connection between the ttt and the excess wealth orders. As two applications of this separation result, we obtain the generating process of the ttt order in terms of mean-decreasing right stretches and the closure property of the ttt order under convolutions. The generating process of the ttt order parallels those of the location independent riskier and the excess wealth orders established by Landsberger and Meilijson (1994).

A characterization of the multivariate excess wealth ordering

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernández-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernández-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.

Stochastic comparisons of distorted variability measures

In this paper, we consider the dispersive order and the excess wealth order to compare the variability of distorted distributions. We know from Sordo (2009a) that the excess wealth order can be characterized in terms of a class of variability measures associated to the tail conditional distribution which includes, as a particular measure, the tail variance. Given that the tail conditional distribution is a particular distorted distribution, a natural question is whether this result can be extended to include other classes of variability measures associated to general distorted distributions. As we show in this paper, the answer is yes, by focusing on distorted distributions associated to concave distortion functions. For distorted distributions associated to more general distortions, the characterizations are stated in terms of the stronger dispersive order.

SOME NEW RESULTS ON ORDERING OF SIMPLE SPACINGS OF GENERALIZED ORDER STATISTICS

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to establish several stochastic comparisons of simple spacings in the mean residual life and the excess wealth orders under the more general assumptions on the parameters of the models.

Comparing tail variabilities of risks by means of the excess wealth order

There is a growing interest in the actuarial community in employing certain tail conditional characteristics as measures of risk, which are informative about the variability of the losses beyond the value-at-risk (one example is the tail conditional variance, introduced by Furman and Landsman ( 2006a, 2006b)). However, comparisons of tail risks based on different measures may not always be consistent. In addition, conclusions based on these conditional characteristics depend on the choice of the tail probability p , so different p ’s also may produce contradictory conclusions. In this note, we suggest comparing tail variabilities of risks by means of the excess wealth order, which makes judgments only if large classes of tail conditional characteristics imply the same conclusion, independently of the choice of p .

On the relationship of location-independent riskier order to the usual stochastic order

Location-independent riskier order and its dual version, excess wealth order, compare random variables in terms of dispersion. In this note, we derive the relationship of both orders to the usual stochastic order. Some new properties of these orders are obtained as a consequence.